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Conductance of 1D quantum wires with anomalous electron-wavefunction localization

We study the statistics of the conductance $g$ through one-dimensional disordered systems where electron wavefunctions decay spatially as $|ψ| \sim \exp (-λr^α)$ for $0 <α<1$, $λ$ being a constant. In contrast to the conventional Anderson localization where $|ψ| \sim \exp (-λr)$ and the conductance statistics is determined by a single parameter: the mean free path, here we show that when the wave function is anomalously localized ($α<1$) the full statistics of the conductance is determined by the average $<\ln g>$ and the power $α$. Our theoretical predictions are verified numerically by using a random hopping tight-binding model at zero energy, where due to the presence of chiral symmetry in the lattice there exists anomalous localization; this case corresponds to the particular value $α=1/2$. To test our theory for other values of $α$, we introduce a statistical model for the random hopping in the tight binding Hamiltonian.